On the complexity of curve fitting algorithms

We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find precise conditions under which this is possible. It turns out that this is, indeed, possible when one fits circles but not ellipses or hyperbolas.Comment: 8 pages, no figure

State space description of national economies: the V4 countries

We present a new approach to description of national economies. For this we use the state space viewpoint, which is used mostly in the theory of dynamical systems and in the control theory. Gross domestic product, inflation, and unemployment rates are taken as state variables. We demonstrate that for the considered period of time the phase trajectory of each of the V4 countries (Slovak Republic, Czech Republic, Hungary, and Poland) lies approximately in one plane, so that the economic development of each country can be assocated with a corresponding plane in the state space. The suggested approach opens a way to a new set of economic indicators (for example, normal vectors of national economies, various plane slopes, 2D angles between the planes corresponding to different economies, etc.). The tool used for computations is orthogonal regression (alias orthogonal distance regression, alias total least squares method), and we also give general arguments for using orthogonal regression instead of the classical regression based on the least squares method. A MATLAB routine for fitting 3D data to lines and planes in 3D is provided.Comment: 13 pages, 18 figure

Fast and numerically stable circle fit

We develop a new algorithm for fitting circles that does not have drawbacks commonly found in existing circle fits. Our fit achieves ultimate accuracy (to machine precision), avoids divergence, and is numerically stable even when fitting circles get arbitrary large. Lastly, our algorithm takes less than 10 iterations to converge, on average.Comment: 16 page

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis of Deeply Virtual Compton Scattering Data at Jefferson Lab and Proton Tomography

The CLAS and Hall A collaborations at Jefferson Laboratory have recently released new results for the ep->ep{\gamma} reaction. We analyze these new data within the Generalized Parton Distribution formalism. Employing a fitter algorithm introduced and used in earlier works, we are able to extract from these data new constraints on the kinematical dependence of three Compton Form Factors. Based on experimental data, we subsequently extract the dependence of the proton charge radius on the quarks longitudinal momentum fraction.Comment: 25 pages, 26 figure

Detection of a Third Planet in the HD 74156 System Using the Hobby-Eberly Telescope

We report the discovery of a third planetary mass companion to the G0 star HD 74156. High precision radial velocity measurements made with the Hobby-Eberly Telescope aided the detection of this object. The best fit triple Keplerian model to all the available velocity data yields an orbital period of 347 days and minimum mass of 0.4 M_Jup for the new planet. We determine revised orbital periods of 51.7 and 2477 days, and minimum masses of 1.9 and 8.0 M_Jup respectively for the previously known planets. Preliminary calculations indicate that the derived orbits are stable, although all three planets have significant orbital eccentricities (e = 0.64, 0.43, and 0.25). With our detection, HD 74156 becomes the eighth normal star known to host three or more planets. Further study of this system's dynamical characteristics will likely give important insight to planet formation and evolutionary processes.Comment: 24 pages, 4 tables, 6 figures. Accepted for publication in ApJ. V2 fixed table 4 page overrun. V3 added reference